Summary: Excellent CC-oriented guide for getting students to adopt the “algebraic” habit of mind with a particularly strong chapter on using puzzles in the classroom.

I’m a first year middle school math teacher trying to broaden my pedagogical understanding of the subject. I have come across many impressive math education books by Jo Boaler, Cathy Seeley, Marilyn Burns and John A. Van de Walle. I’ve also picked a few recent titles which are “Common Core” aware (such as Cathy Humphreys’ Making Number Talks Matter, Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris and finally Making Sense of Algebra by E. Paul Goldenberg and others). All are excellent in their own way. “Making Sense of Algebra” selects a small number of topics and covers them in depth; the problems and puzzles it presents would fit perfectly well in high school algebra as well as a class for advanced middle school students. At the same time, the book covers some fundamental topics which properly should be taught at the middle school level (or earlier).

Making Sense of Algebra does not contain lesson plans or activity worksheets. While the book alludes frequently to CC math standards, it doesn’t try to review these standards or at least provide a reference to them (that might have helped). Although the book has multiple names in the byline, it has a good logical flow and certainly doesn’t read like an education textbook (it’s much better!) With an important exception noted below, the book doesn’t really cover geometry, nor does it refer to trigonometry or calculus in any in-depth way. Still, the general principles of solving math problems elucidated here do apply to all kinds of higher math.

Rather than trying to plan a class or curriculum, the book covers the development of mathematical habits of mind.

The first chapter introduces the concept of “algebraic habits of mind” and how it relates to the Common Core’s Standards for mathematical practice. Chapter 2 discusses problems in contemporary math education and the special challenges facing certain kinds of struggling learners. Chapter 3 covers how puzzles can be used in class to promote algebraic habits of mind. Chapter 4 talks about how teachers can help students to investigate problems and formulate solutions. Chapter 5 talks about the importance of revising certain mental models commonly used in lower grades to illustrate multiplication and negative numbers. It shows why using number lines to illustrate addition and subtraction obviate the need to teach certain rote rules (like “multiplying two negatives cancels each other out”) and that using the metaphor of area to illustrate multiplication lays the groundwork for explaining how to multiply polynomials.The last chapter covers how a teacher can monitor and tighten language used in the classroom to best facilitate learning. It also provides insights into how a teacher can overcome a student’s reluctance to talk in math class.

I found the chapter on puzzles to be the most remarkable and helpful to me as a teacher. It can be a challenge though to use them in class. Some puzzles that are too hard (or too dependent on non-mathematical skills) can end up segregating the class into those willing to try hard puzzles and those who don’t even bother. For example, I — like many other math teachers — introduced the infamous Cheryl’s birthday math problem to my middle school students. My top students found it challenging but perplexing while a good chunk of my students didn’t even try (despite some pre-teaching about how to systematically record guesses, etc). The puzzle chapter makes a case about the pedagogical value of having students experience frustration and try a variety of approaches to solve something. It covers lots of different puzzle types which are more specifically about math (unlike the Cheryl’s birthday problem), more inviting to students and apt to lead students down algebraic paths. The book discusses the learning opportunities of various puzzle types and the advantage of using puzzle types which are easy for a teacher (or student) to create on their own. The idea of students creating math puzzles was intriguing to me, but it makes perfect sense; it helps students with “posing interesting problems” which is another habit of mind which the book believes to be important.

The book suggests that puzzles be used as “stand-alone investigations” rather than introducing them during units when a specific topic is studied. The book defends this practice by saying: “Life’s real problems arrive at any time, not just when you are conveniently studying how to solve them. We investigate when we don’t know how to solve a problem. We must not start out by thinking, ‘Oh, I’m supposed to factor because that’s what we’re studying now.'” The book argues that cultivation of “stamina” is important when when trying to solve math problems and that “problems which are too short or too scaffolded don’t increase students’ investigation skills or stamina.” For this reason, it’s helpful to give students problems with a “low threshold, high ceiling” (translation: problems which are easy to play with, but might involve concepts beyond their zone of proximal development).

The book offers several strategies for helping to cultivate student’s investigative skills. First, it emphasizes the importance of gaining experience about the problem itself before trying to formalize a solution. This can involve plugging in a few haphazard numbers or using experimental aids. Second, the teacher can give “tail-less” or “headless” problems whereby students are given a set of facts without an actual question being asked and must write a list of assumptions implied by this set of facts (or conversely, the student is given a problem and asked to speculate about what data is needed to solve it). What a good idea! Often failing to recognize the implications of a mathematical statement can prevent the student from reaching a solution. Third, presenting students with redundant quantitative information in a problem can make it easier for struggling students to make connections. Fourth, providing additional questions (i.e., “have you found ALL the solutions?”) can be a challenging and interesting way to extend the assignment for advanced students.

While the first half of the book did a great job of explaining how students think mathematically and how to make them think more productively, I was beginning to think that the book offered little real insight about how to run a math class and organize students effectively. Some questions spring to mind: 1)how do you do assessments of puzzle solving or habits of mind? 2)what kinds of topics lend themselves better to small group activities and what kinds require more teacher-prodding? 3)How do you integrate the need to teach habits of mind with the need to teach mandated objectives?

The second half of the book tackles these kinds of questions. The investigations chapter ends with a fairly good discussion of how to structure whole class discussions of investigations after students have collaborated on clarifying examples. The subject of the last chapter “Thinking Out Loud” is about the best ways how to teach students to discuss mathematical ideas in the classroom. The book stresses the importance of encouraging students to “think first, then talk,” but argues that discussions are a way to “vary the texture of the class.” I recently finished Cathy Humphreys Making Number Talks Matter and feel that this book provides a exhaustive treatment of the value of a more communicative approach to math and how to implement it. The chapter in Thinking Algebraically covers some of the same ground (without as many examples), but it makes several important new points. First, the teacher should encourage and model precision in speech. For example, when discussing a cube, using the word “side” invites misunderstanding; if you use “vertices,” “edges”, “faces”, that reduces the possibility of confusion. (Of course, it is impossible for students to avoid using “sloppy” language, but it is possible to make students aware of the need for precision). It’s important to choose topics which are actually discussable and to give the student enough time to formulate an answer (the book says “counting to 20 in your head….is not unreasonable”).

The book analyzes in great detail the various reasons why students prefer not to talk in a math class. Perhaps the question seems too trivial, or the student may lack confidence in their own math skills to express their ideas. It offers ways that teachers can encourage productive discussions. For example, instead of saying “close” or “you are getting warmer,” the teacher can respond to a wrong answer with supportive statements like “the answer needs to be even” or “were you thinking that 7×7= 49?” The book offers ways for the teacher to make the student feel empowered in the classroom and links the ability to solve puzzles appropriate to their level as a confidence-builder. One recommended technique is to present written fictional “math dialogues” about a math situation, and have students read along and critique the approaches of the fictional students. Although these dialogues may sound corny, “the student reading it can imagine — even without knowing this is fiction — how characters who are never told what to do or how to do it can believe and demonstrate that they can figure out mathematical ideas for themselves using what they already know. This invests mathematical authority in these characters, repeatedly giving the message that mathematical knowledge can be built logically rather than from some external source.”

This is a brilliant insight and a great way to model student conversations and habits of mind. The book provides one extended example of a fictional dialogue and references to other books which contain additional dialogues. (I would have liked the book to have a second example, but this is fine).

My only complaint is that I wish the book had covered how technology and videogames can be incorporated in class. In Texas, all middle schoolers are expected to follow self-guided online lessons and videogames called Think Through Math. I have recently been wowwed by the Dragonbox Algebra 12+ mobile app/game (described in detail in Greg Toppo’s book The Game Believes in You). For various organizational and budgetary reasons, math departments are having to use these kinds of courses and modules, and teachers could benefit from guidance about whether these methods can be academically rigorous and easily integrated into the classroom. I suspect that the book’s authors would be skeptical of algebra via videogames. At the same time, students have lots of access to math resources via the web; are these “cheats” pedagogically useful? Or should the teacher make some attempt to discourage students from finding the answer online so they may arrive at their own insights?

OVERALL this compact book is a pretty dense read, but full of insights and really fun to read. (I enjoyed trying out many of the puzzles myself). This book showed an awareness of existing scholarship and provided an ample bibliography, making it invaluable for the novice teacher (though the experienced math teacher will find useful insights here as well). I fear that the book will be known mainly for exploring the use of puzzles in the classroom. But the book covers a lot more ground than that.